I think it's cute, but my first thought was the word 'via' and I think that could cause confusion and mispronunciation. but as a nn for any of the suggested names I love it. Some other options you could consider...

I'm in the same boat as you. I adore the name Livy and Liv, and I'm thinking of side stepping uber popular Olivia as well.
But I agree with the others that it feels very much like a nickname and all I hear is "through" or "way" when I hear Via. It would be akin to using Versa.